Question

Find the locus of a complex number, z = x + iy, satisfying the relation `|[ z -3i}/{z +3i]| ≤ sqrt2 `. Illustrate the locus of z in the Argand plane.

Answer 1

`|[ z -3i]/[z +3i]| ≤ sqrt2 `

⇒ `|[x + iy - 3i]/[x +iy +3i]| ≤ sqrt2 `

⇒ `|x + i (y - 3)| ≤ sqrt2   |x + i (y + 3)|`

⇒ `sqrt(x^2 + (y - 3)^2) ≤ sqrt2  sqrt(x^2 + (y + 3)^2)`

⇒  x² + (y - 3)² ≤ 2  (x² + (y + 3)²)

⇒ x² + y² + 9 - 6y ≤ 2x² + 2(y² + 9 + 6y)

⇒ 2x² + 2y² + 18 + 12y - x² - y² - 9 + 6y ≥ 0

⇒ x² + y² + 9 + 18y ≥ 0

⇒ x² + y² + 18y + 9 + 81 ≥ 81

⇒ x²  + (y + 9)² ≥ 72

⇒ (x - 0)² + (y - (-9))² ≥ ( 6√2)²

This represents a circle with centre (0, 9) and radius 6√2 units.